Malcolm Tukcer doesn't plan on stopping in Mayberry but a busted fuel line finds him wandering through the empty and quiet streets of Mayberry on a Sunday afternoon. After locating Andy, leaving church with his family, he explains his predicament. He has to be in Charlotte Monday morning for urgent business and must have his car repaired as quickly as possible. Despite Andy explaining that very little happens in Mayberry on a Sunday outside of going to church, Mr. Tucker insists on being taken to Wally's filling station.
Andy and Mr. Tucker arrive at Wally's to find that station barely open, as Andy puts in. Gomer Pyle makes his first official appearance in Mayberry as does Wally's filling station. Although Gomer is on duty, all he can do is pump gas, Wally is Mayberry's only mechanic.
Mr. Tucker and Andy are off to Wally's. Wally is quickly able to diagnose the root cause of Mr. Tucker's ailing automobile but insists that he will do no work on Sunday! An irritated Mr. Tucker storms off, determined to find some way to Charlotte! He goes back to Wally's filling station and questions Gomer, knowing that someone in Mayberry must know how to fix cars other than Wally. After much thought, Gomer realizes his cousin Goober (the first mention of the other Pyle cousin would would eventually appear in season five to take Gomer's place at Wally's when Gomer joins the Marines) can fix Mr. Tuckers car only Goober is not at home, he's out on his boat (according to Gomer he hopped up an old V-8 and put it on his row boat). Distraught, Mr. Tucker steals Gomer's truck and speeds off, bound to find some way out of Mayberry! He is picked up and brought back to town by Andy and Gomer's truck is returned. Rather than arrest and charge Mr. Tucker, Andy lets the matter go and invites Mr. Tucker home for dinner.
Later at the Taylor home, Andy, Opie, and Barney are getting ready for Sunday dinner. Mr. Tucker, still insistent on getting out of Mayberry attempts to make a phone call only to find that Mayberry's only long distance line is use by the Mendlebright sisters who spend every Sunday afternoon visiting. Mr. Tucker slams the phone down, amazed that in a world with international TV and space flight, a whole town can shut down.
A little while later, while sitting on the front porch, Gomer shows up. Goober has shown up, they've towed Mr. Tucker's car in and Goober is at that very minute working on the clogged fuel line. A while later Gomer drives up in Mr. Tucker's car. After refusing payment, Mr. Tucker prepares to finally leave Mayberry. However, as he's leaving, he realizes what he's been missing and that despite the setbacks, the slow and easy pace of Mayberry is quite appealing to him. He makes the decision to stay the night and finds out the his urgent business is not so urgent after all.
The final shot is of the Taylor front porch at twilight. Mr. Tucker is sitting in Aunt Bee's rocking chair asleep, a half peeled apple in his hand.